
By marginalizing over the model parameters \(t\)  and \(\alpha \) , we obtain in Sec.  the prior predictive distributions \(P(I|{\cal H}_i)\)  for each hypothesis. These inform us on which vacua are statistically favored, without taking our data into consideration. They also allow us to compute probability distributions for various physical parameters, in particular the cosmological constant (CC).

The uniform-time and late-time hypotheses will compete in Sec.  by computing the posterior odds ratio \(\frac{P({\cal H}_{\rm late}|D)}{P({\cal H}_{\rm uni}|D)}\) . Assuming comparable prior odds for the two hypotheses, we will find, under general and reasonable assumptions about transition rates, that posterior odds exponentially favor the uniform-time hypothesis. That is, the GSVW measure [1]} is exponentially disfavored compared to the holographic prior probabilities [2]}. This is a key result of our analysis. There are possible caveats and loopholes in our analysis, and we will carefully spell these out in Sec. REF .

Conditioning on our data \(D\) , we will perform parameter inference in Sec. . We will be specifically interested in the posterior probability distribution \(P(t|D,{\cal H}_i)\)  for the time of nucleation. Focusing on \({\cal H}_{\rm uni}\) , we will find that the average time for occupying vacua compatible with our data is much shorter than mixing time, confirming the assumptions underlying the early-time approach [3]}, [4]}, [5]}, [6]}, [7]}.

